The Resultant of Two Vectors

The Resultant of Two Vectors — Moosa Academy

When two vectors act at once, a single vector can replace them both. That replacement is called the resultant, and there are two standard ways to draw it — the triangle rule and the parallelogram rule.

Concept What a resultant is

The resultant of $\vec{a}$ and $\vec{b}$ is the single vector $\vec{a} + \vec{b}$ that produces the same overall effect as the two together.

Both construction methods below give the same resultant. They are two ways of drawing one idea, not two different answers.

Theorem The triangle rule

Slide

\vec{b}

so that its tail sits on the tip of

\vec{a}

. The resultant runs from the tail of a to the tip of b, closing the triangle.

Sliding a vector does not change it — only its position on the page moves, while its length and direction stay the same.

Theorem The parallelogram rule

Draw

\vec{a}

and

\vec{b}

from the same starting point, complete the parallelogram, and the diagonal from that shared point is the resultant.

Example Using the triangle rule

Two vectors $\vec{a}$ and $\vec{b}$ are given. Find $\vec{a} + \vec{b}$ .

Draw

\vec{a}

in its given position.

Slide

\vec{b}

so it begins at the tip of

\vec{a}

Join the tail of

\vec{a}

to the tip of

\vec{b}

⟹ that closing arrow is the resultant

Example Using the parallelogram rule

The same two vectors, now drawn from a common point.

Place

\vec{a}

and

\vec{b}

with their tails together.

Draw the two parallel sides to complete the parallelogram.

Draw the diagonal from the shared starting point.

⟹ the same resultant as before

Use whichever suits the diagram: the triangle rule chains vectors head to tail, while the parallelogram rule keeps both starting from one point.

Summary

The resultant is one vector with the same effect as two acting together.
Triangle rule: place b's tail at a's tip; the resultant closes the triangle.
Parallelogram rule: draw both from one point; the resultant is the diagonal.
Sliding a vector changes its position but not its length or direction.
Both rules always produce the same resultant.